The state might require that to avoid liability a private association must maintain records of written ackowledgements from each of its members indicating that they are choosing to associate with an organization that retains the option of discriminating on certain bases.
A lot of people couldn't bring themselves to sign such a document.
Question number one: is it constitutional? I'm skeptical--it seems that it would be seen as an attempt to intimidate unpopular associative groups. Such actions are inherently suspect (see NAACP v. Alabama ex rel. Patterson).
Question number two: is it a desirable precedent? Regardless of whether we can do this, I'm not sure we should. I'm not sure we shouldn't, either, but it seems to me there have to be some repercussions out there if liberals suddenly begin supporting requiring associations to publicly recognize when they differ with public policy. Imagine if the law schools win the Solomon Case currently before the Supreme Court. Would this then mean that Congress could require all affiliates of that university to sign a waiver saying, "I am joining an organization knowing that it has gained a legal exception and is allowed to discriminate against and exclude the armed forces from recruiting on its campus"? It seems the answer is yes. I'm uncomfortable with setting the precedent that the government can force certain speakers/groups to label themselves as "law-exempted" (implying, of course, that were it not for that pesky Supreme Court we would throw them all in jail).
It's an interesting idea, but a dangerous one.